RATIO RATES AND PROPORTIONS

Using Ratios

Example 1 :

The ratio of faculty members to students at a college is 1:15. There are 750 students. How many faculty members are there?

Solution :

Write a ratio comparing faculty to students.

Faculty ----> 1

Students ----> 15

Write a proportion. Let x be the number of faculty members.

ˣ⁄₇₅₀ = ¹⁄₁₅

Because x is divided by 750, multiply each side of the equation by 750.

750 ⋅ (ˣ⁄₇₅₀) = (¹⁄₁₅) ⋅ 750

x = 50

There are 45 faculty members.

Finding Unit Rates

Example 2 :

David ate 58.5 hot dogs in 13 minutes to win a contest. Find the unit rate in hot dogs per minute. Round to the nearest hundredth.

Solution :

Write a proportion to find an equivalent ratio with a second quantity of 1.

⁵⁸^.⁵⁄₁₃ = ˣ⁄₁

4.5 = x

The unit rate is approximately 4.46 hot dogs per minute.

Using Dimensional Analysis

Example 3 :

A large adult male human has about 12 pints of blood. Use dimensional analysis to convert this quantity to gallons.

Solution :

Step 1 :

Convert pints to quarts.

Multiply by a conversion factor whose first quantity is quarts and whose second quantity is pints.

12 pt ⋅ (1 qt/2 pt) = 6 qt

12 pints is 6 quarts.

Step 2 :

Convert quarts to gallons.

Multiply by a conversion factor whose first quantity is gallons and whose second quantity is quarts.

6 qt ⋅ (1 gal/4 qt) = 6/4 gal = 1½ gal

A large adult male human has about 1½ gallons of blood.

Example 4 :

The dwarf sea horse Hippocampus zosterae swims at a rate of 52.68 feet per hour. Use dimensional analysis to convert this speed to inches per minute.

Solution :

1 foot = 12 inches

1 hour = 60 minutes

Use the above conversion factors to convert feet to inches and hours to minutes.

52.68 ft ⋅ 12 in. = 632.16 in.

1 hr ⋅ 60 min = 60 min

Then,

The speed is 10.536 inches per minute.

Solving Proportions

Solve each proportion.

Example 5 :

⁷⁄₉ = ³⁄ₓ

Solution :

⁷⁄₉ = ³⁄ₓ

Use cross products.

7x = 3(9)

7x = 27

Divide each side by 7.

x = 27/7

Example 6 :

⁵⁄₉ = ¹⁰⁄₍ₓ ₊ ₂₎

Solution :

⁵⁄₉ = ¹⁰⁄₍ₓ ₊ ₂₎

Use cross products.

5(x + 2) = 9(10)

5x + 10 = 90

Subtract 10 from each side.

5x = 80

Divide each side by 5.

x = 16

Example 7 :

On a map with a scale of 1 in = 18 mi, the distance from Chicago to Evanston is 0.63 in. What is the actual distance ?

Solution :

Write the scale as a fraction.

map/actual = 1 in/18 mi

Let x be the actual distance.

¹⁄₁₈ = ⁰^.⁶³⁄ₓ

Use cross products to solve.

1(x) = 18(0.63)

x = 11.34 miles

The actual distance is 11.34 miles.

Example  8 :

The actual distance between the two cities A and B is 54 mi. Find this distance on a map with a scale of 1 in = 18 mi. Round to the nearest tenth.

Solution :

Write the scale as a fraction.

map/actual = 1 in/ 18 mi

Let x be the distance on a map.

ˣ⁄₅₄ = ¹⁄₁₈

Use cross products to solve.

18(x) = 54(1)

18x = 54

Divide each side by 18.

18x/18 = 54/18

x = 3

The distance on the map is 3 inches.

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